Color-measurement instruments (e.g., spectrophotometers) can be characterized parametrically and corrected to measure like a reference or master instrument. The characterization and correction is based on instrument profiles. The “profile” of a first instrument relative to a second instrument is defined herein as a set of parameter values that mathematically map a first set of reflectance values of a set of specimens as measured by the first instrument to the best approximation of a second set of reflectance values of the same specimens as measured by the second instrument. The act of “profiling” is defined herein as the generation of these parameter values, and “profile-based correction” is defined herein as the act of implementing the mapping defined by these parameter values to correct subsequent reflectance measurements by the profiled instrument. Such profile-based correction compensates for small systematic differences between instruments. Profile-based correction typically starts from a model equation such as the following:Rci=A+BRmi+CR′miDR″mi−+ERmi(100−Rmi)  (EQN. 1)Where Rmi is the ith measured reflectance, Rci is the ith corrected reflectance, all variables A, B, C, D, and E implicitly depend on wavelength, and ′ and ″ refer to the first and second derivatives, respectively, of Rmi with respect to wavelength. The corrections in EQN. 1 are represented by the following parameters: offset (A), gain change (B), wavelength-scale change (C), bandwidth change (D), and some nonlinearities (E). EQN. 1 (or a similar equation, possibly with a different number of parameters) is used twice in profile-based correction. The first usage of EQN. 1 measures known specimens such as the British Ceramic Research Association (BCRA) tiles with parameters A-E (at each wavelength) in the solve state. In this first usage of EQN. 1, optimizing software is typically used to find the parameter values A-E that make the computed quantities Rci closest to the quantities Rmi of the second instrument. The now-known quantities A-E (and possibly other parameters) comprise the profile of the first instrument relative to the second instrument. During the second usage of EQN. 1, the first instrument measures reflectances of test specimens, adopts the measured reflectances of the test specimens as the quantities Rmi, substitutes the now-known parameters A-E into EQN. 1, and uses EQN. 1 to compute the corrected measurements Rd. Continued operation of the second usage of EQN. 1 constitutes the profile-based correction of the reflectance measurements.
Profile-based correction of a fleet of spectrophotometers depends on the availability of a master instrument (to act as the second instrument described above) and at least one set of trustworthy color standards (i.e., reflecting specimens such as the BCRA tiles). Such correction also works best when the master instrument is very close in design to the instruments that are being corrected to it in the fleet. In some cases, the trusted master instrument may have a slightly different design than the instruments in the fleet; however, the master instrument must still be used as an initial anchor if no better standard exists. In such a case, one can improve the inter-instrument agreement by computing the average reflectance data from the fleet of instruments and correcting the fleet of instruments to this average. However, this introduces another problem: the use of the average reflectance data requires use of the same reflecting color standards for all of the profiling measurements. This is impractical, especially when the instruments in the fleet are not geographically co-located. One must have at hand either a real master instrument or a real set of standard color tiles that is used for all corrections.